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Numerical analysis of surface waves on permeable boundaries of poroelastic media

Collaborator: B. Albers

Cooperation with: C. Lai (European Centre for Training and Research in Earthquake Engineering, Pavia, Italy), R. Lancellotta, S. Foti (Politecnico di Torino, Italy)

Description:

In the year 2004, the work of the last years on the numerical analysis of surface waves in poroelastic media (e.g., [4], [3]) was confirmed. The theoretical research on surface waves is based on the ``simple mixture model'' by Wilmanski [6].

In 2004, the boundary between a porous medium and an ideal fluid has been investigated. This means that there is one additional component compared to the boundary porous medium/vacuum which has been investigated in 2003. Thus, besides the three bulk waves in the porous medium, there exists also a P wave in the fluid. These four waves combine into three surface waves: a leaky Rayleigh wave and both a true and a leaky Stoneley wave. Their acoustic properties (phase and group velocities, attenuations) are shown in the papers [2], [1] in dependence on two quantities: the frequency $\omega$ and the surface permeability parameter $\alpha$ (see figure). The variation of the second parameter, $\alpha$ , which controls the intensity of the in- and outflow of the fluid from the porous medium, brought to light that the true Stoneley wave exists only for very small values of this parameter, i.e. for a boundary which is almost sealed. Attenuations of both leaky waves show an interesting behavior in dependence on the frequency: for two frequencies there appear resonance effects (see figure). They seem to be not only theoretically but also experimentally observed, [5], and may be related to characteristic frequencies of the solid and the fluid, respectively.

Summary of results

The three observed surface waves possess the following attributes:

Leaky Rayleigh

- The velocity of propagation of this wave lies in the interval determined by the limits $\omega$ $\rightarrow$ 0 and $\omega$ $\rightarrow$ $\infty$ . The high frequency limit is higher than the low frequency limit. The velocity is always smaller than c_S, i.e. slower than the S wave. As a function of $\omega$ , it possesses at least one inflection point.

- For low frequencies, the phase velocity for different values of the surface permeability $\alpha$ remains almost constant. For high frequencies, smaller values of $\alpha$ yield larger velocities; for the open pore case, the difference between high and low frequency limits is approximately one half of the difference for a close boundary.

- The attenuation grows linearly and unboundedly (the feature of a leaky wave), there appear singularities which depend on $\alpha$ and seem to be related to the characteristic frequencies ${\frac{{\pi }}{{2\rho _{0}^{S}}}}$ and ${\frac{{\pi }}{{2\rho _{0}^{F}}}}$ .

Leaky Stoneley

- The phase velocity of this wave behaves similarly to the one of the leaky Rayleigh wave. However, the high frequency limit is larger for bigger values of $\alpha$ than for smaller ones; a maximum value appears in the region of order 100 kHz. The velocity of the leaky Stoneley wave is for each pair $\left(\vphantom{ \omega,\alpha }\right.$ $\omega$ , $\alpha$ $\left.\vphantom{ \omega,\alpha }\right)$ smaller than the one of the leaky Rayleigh wave.

- Also the attenuation behaves similarly to the one of the leaky Rayleigh wave. However, the singularities are more weakly dependent on $\alpha$ .

True Stoneley

- It exists only for small values of the surface permeability $\alpha$ . For different values of $\alpha$ , the velocity is nearly the same. It grows monotonically from the zero value for $\omega$ = 0 to a finite limit which is slightly smaller (approximately 0.15 %) than the velocity c_P2 of the P2 wave. The growth of the velocity of this wave in the range of low frequencies is much steeper than the one of Rayleigh waves similarly to the growth of the P2 velocity.

- Both the velocity and attenuation of the true Stoneley wave approach zero as $\sqrt{{\omega }}$ (which is not directly obvious due to the logarithmic scale of the figures).

- The attenuation of the Stoneley wave grows monotonically to a finite limit for $\omega$ $\rightarrow$ $\infty$ . It is slightly smaller than the attenuation of P2 waves.

References:

B. ALBERS, Monochromatic surface waves at the interface between poroelastic and fluid halfspaces, in preparation.
, Modelling of surface waves in poroelastic saturated materials by means of a two component continuum, WIAS Preprint no. 952, 2004 , to appear in: Surface Waves in Geomechanics, Direct and Inverse Modelling for Soils and Rocks, C. Lai, K. Wilmanski, eds., CISM Courses and Lectures, Springer, Wien, New York.
, Numerical analysis of monochromatic surface waves in a poroelastic medium, WIAS Preprint no. 949, 2004 , submitted.
B. ALBERS, K. WILMANSKI, Monochromatic surface waves on impermeable boundaries in two-component poroelastic media, submitted.
G. CHAO, D.M.J. SMEULDERS, M.E.H. VAN DONGEN, Shock induced borehole waves in porous formations: Theory and experiments, J. Acoust. Soc. Am., 116 (2004), pp. 693-702.
K. WILMANSKI, Waves in porous and granular materials, in: Kinetic and Continuum Theories of Granular and Porous Media, K. Hutter, K. Wilmanski, eds., vol. 400 of CISM Courses and Lectures, Springer, Wien, New York, 1999, pp. 131-186.

**Fig. 1:** Leaky Rayleigh
$\makeatletter \@ZweiProjektbilderNocap[h]{0.38\textwidth}{Rayleigh.ps}{Rayleighatt.ps} \makeatother$

**Fig. 2:** Leaky Stoneley
$\makeatletter \@ZweiProjektbilderNocap[h]{0.38\textwidth}{Stoneleyaussen.ps}{Stoneleyaussenatt.ps} \makeatother$

**Fig. 3:** Stoneley
$\makeatletter \@ZweiProjektbilderNocap[h]{0.38\textwidth}{Stoneley.ps}{Stoneleyatt.ps} \makeatother$

Normalized phase velocities and attenuations of the leaky Rayleigh, leaky Stoneley, and the Stoneley wave in dependence on the frequency. Different curves correspond to different values of the surface permeability $\alpha$ (in units $\left[\vphantom{\frac{\mathrm{s}}{\mathrm{m}}}\right.$ ${\frac{{\mathrm{s}}}{{\mathrm{m}}}}$ $\left.\vphantom{\frac{\mathrm{s}}{\mathrm{m}}}\right]$ ).